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Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Royal Institute of Technology.
2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed) Published
Abstract [en]

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

Place, publisher, year, edition, pages
Amsterdam: Elsevier, 2018. Vol. 330, p. 177-192
Keywords [en]
Nonlinear eigenvalue problems, Helmholtz problem, Scattering resonances, Dirichlet-to-Neumann map, Arnoldi's method, Matrix functions
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-138325DOI: 10.1016/j.cam.2017.08.012ISI: 000415783000014OAI: oai:DiVA.org:umu-138325DiVA, id: diva2:1134522
Available from: 2017-08-21 Created: 2017-08-21 Last updated: 2019-05-20Bibliographically approved
In thesis
1. Reliable hp finite element computations of scattering resonances in nano optics
Open this publication in new window or tab >>Reliable hp finite element computations of scattering resonances in nano optics
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2019. p. 35
Series
Research report in mathematics, ISSN 1653-0810 ; 67
Keywords
Scattering resonances, Helmholtz problems, pseudospectrum, Lippmann-Schwinger equation, finite element methods, nonlinear eigenvalue problems, spurious solutions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159154 (URN)978-91-7855-076-0 (ISBN)
Public defence
2019-06-13, MA121, MIT-huset, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-05-23 Created: 2019-05-20 Last updated: 2019-05-21Bibliographically approved

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Araujo-Cabarcas, Juan CarlosEngström, Christian

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